9 THE PLANCHEREL-GODEMENT THEOREM 371 A. Show that there exists a decomposition of H as a Hilbert sum of a subspace H0 and a (finite or infinite) sequence of closed subspaces H* stable with respect to B, with the following properties: (1) the restrictions to H0 of the operators belonging to B are all zero; (2) each Hfc is the Hilbert sum of a finite sequence (Hfcp)i <p<rfc of subspaces of the same dimension (finite or not) which are stable with respect to B; the restrictions to Hfci of the operators belonging to B form the algebra of Hilbert-Schmidt operators on Hki; moreover, for 2^p^ rk, there exists an isometric isomorphism Tp of HfcJ. onto Hfcp such that, if Ui is the restriction to Hki of an operator U e B, then the restriction of U to Hfcp is TpUiTp1. 9. THE PLANCHEREL-GODEMENT THEOREM In this section we shall study commutative (but not necessarily complete) Hilbert algebras. More generally it will be convenient, in view of applications, to consider a commutative algebra A with involution, endowed with a bitrace g (15.7) satisfying conditions (U) and (N) of (15.6). We shall denote by n^ the two-sided ideal of elements s e A such that g(s, s) = 0, and by ng : A -*• A/ng the canonical mapping. We recall that A/ng is canonically endowed with a structure of Hilbert algebra (15.7). We shall assume throughout that the prehilbert space A/ng is separable, and hence is a dense subspace of a separ- able Hilbert space, which we denote by Hg. Thus, starting from g, we obtain canonically a representation of A in H^ (15.6.10), which we denote by Ug. The image of A under Ug is a commutative subalgebra with involution (15.6.1) of £?(Hg). Let ^denote its closure in &(Hg), so thatsfgis a com- mutative star algebra (and therefore consists of normal operators (15.4.11)). We shall assume that this algebra jfg is separable (this is not a consequence of the separability of A/ng, cf. Problem 1). A particular example of a trace on A (which therefore gives rise to a bitrace, by the canonical procedure (15.6.2)) is provided by the hermitian characters of A, i.e., the characters % of A which satisfy (15.9.1) *(**) = Xto for all x e A. It follows that x(x*jc) = I x(x)\2 ^ 0, and if we put g(x, y) = i(y*x) = x(y)x(x)> the condition (U) of (15.6) is trivially verified, because g(st, st) = \X(s)\2 \x(t)\2 = \x(s)\2ff(t, t\ On the other hand, the ideal n^, which is here the kernel of #, is a hyperplane in A, so that the algebra A/ng can be identified with C, and the condition (N)Banach